ELLIPTIC K3 SURFACES
DIMA AL TABBAA AND ALESSANDRA SARTI
Abstract. In this paper we classify complexK3 surfaces with non-symplectic automorphism of order 8 that leaves invariant a smooth elliptic curve. We show that the rank of the Picard group is either 10, 14 or 18 and the fixed locus is the disjoint union of elliptic curves, rational curves and points, whose number does not exceed 1, 2, respectively 14. We give examples corresponding to several types of fixed locus in the classification.
Introduction
The study of automorphisms of K3 surfaces started with the pioneering work of Nikulin [11] and several people contributed then to the developpement of the theory, see [17] for a short survey. In the paper we investigate purely non-symplectic automorphisms of order 8, i.e. automorphisms that multiply the non-degenerate holomorphic two form by a primitive 8th root of unity.
The study of non-symplectic automorphisms of prime order was completed by several authors: Nikulin in [12], Artebani, Sarti and Taki in [3, 5, 14]. The study of non-symplectic automorphisms of non-prime order turns out to be more compli- cated. Indeed, in this situation the ”generic” case does not imply that the action of the automorphism is trivial on the Picard group [8, Section 11]. In the pa- per [15], Taki completely describes the case when the order of the automorphism is a prime power and the action is trivial on the Picard group. If we consider non- symplectic, non-trivial automorphisms of order 2t, then by results of Nikulin we have 1≤t ≤5, and by a recent paper by Taki [16] there is only one K3 surface that admits a non-symplectic automorphism of order 32. Some further results in this direction are contained in a paper by Sch¨utt in the case of automorphisms of a 2-power order [13] and in a paper by Artebani and Sarti in the case of order 4 [4]. Recently in [2] the two authors and Taki completed the study for purely non-symplectic automorphisms of order 16.
This paper mainly deals with purely non-symplectic automorphismsσof order eight on elliptic K3 surfaces under the assumption that thier fourth power σ4 is the identity on the Picard lattice. This corresponds to the situation for the generic K3 surface in the moduli space of K3 surfaces with non-symplectic automorphism of order 8 and fixed action on the second cohomology with integer coefficients, see [8, Section 10]. The fixed locus Fix(σ) of such an automorphismσis the disjoint union of smooth curves and points. In the paper we give a complete classification in the case that Fix(σ4) contains an elliptic curve. More precisely let X be a K3 surface,ωX a generator ofH2,0(X),σ∈Aut(X) such thatσ∗ωX =ζ8ωX, whereζ8
2010Mathematics Subject Classification. Primary 14J28; Secondary 14J50, 14J10.
Key words and phrases. non-symplectic automorphisms, K3 surfaces.
1
denotes a primitive 8th root of unity. We denote byr, l, mandm1 the rank of the eigenspaces ofσ∗inH2(X,C) relative to the eigenvalues 1,−1, iandζ8respectively.
We also denote by k the number of smooth rational curves fixed by σ, by N the number of isolated fixed points in Fix(σ). We prove the following result:
Theorem 0.1. Let σ be a purely non-symplectic order eight automorphism acting on a K3 surfaceX such that σ4 acts identically on Pic(X). Assume thatFix(σ4) contains a curveC of genus g(C) = 1, then we have the following possibilities for the fixed locus
• If Fix(σ) contains the elliptic curveC, then (k, N,rk Pic(X)) = (0,2,10),(0,4,14).
• If Fix(σ2) contains the elliptic curveC and there is no elliptic curve fixed byσ, then
(k, N,rk Pic(X)) = (0,2,10), (0,6,10), (0,4,14), (1,10,14).
• If σi fori= 1,2 does not fix an elliptic curve, then
(k, N,rk Pic(X)) = (0,2,10), (0,4,14), (0,2,14), (0,6,14), (1,8,14), (0,2,18), (0,6,18),(2,14,18).
In Table 5 we list all the possibilities in detail and we describe the action of σ on the fibers of the elliptic fibration. We have in total 16 possibilities and we give examples for all the cases except for some special cases when the action ofσ is a translation on the invariant elliptic curve or it acts as a rotation on some reducible fibers of the fibration.
The results of the paper are partially contained in the PhD thesis [1] of the first author under the supervision of the second author.
Acknowledgements: This work was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. Part of the work was done by the second author during the Polish Algebraic Geometry Mini-Semester, 18 april-18 june 2016. She warmly thanks the organizers for the stimulating working atmosphere.
1. Basic facts
LetX be aK3 surface andσ∈Aut(X). We assume thatσ∗ωX =ζ8ωX, where ζ8 = e2πi8 is a primitive 8th root of the unity. Such a σ is called purely non- symplectic, for simplicity we just call it non-symplectic, always meaning that the action is by a primitive 8th root of unity.
We denote byrσj, lσj, mσj and m1 for j = 1,2,4 the rank of the eigenspace of (σj)∗ in H2(X,C) relative to the eigenvalues 1,−1, i and ζ8 respectively (clearly mσ4 = 0). For simplicity we just write r, l, m forj = 1. We recall the invariant lattice:
S(σj) ={x∈H2(X,Z)|(σj)∗(x) =x}, and its orthogonal complement:
T(σj) =S(σj)⊥∩H2(X,Z).
Since the automorphism acts purely non-symplectically,X is projective, see [11, Theorem 3.1]) so that if we denote rkS(σj) = rσj, we have that rσj > 0 for all j = 1,2,4 (one can always find an invariant ample class). On the other hand, one
can easily show thatS(σj)⊆Pic(X) forj= 1,2,4 so that the transcendental lattice satisfiesTX ⊆T(σj) forj= 1,2,4. For simplicity we writeT(σ) :=T(σ1). Recall that the action ofσonTX is by primitive 8th roots of the unity, see [11, Theorem 3.1 (c)].
Remark 1.1. It is a straightforward computation to show that the invariants rσj, lσj, mσj andm1 with j= 1,2,4satisfy the following relations:
rσ2 =r+l; rσ4=r+l+ 2m;
lσ2 = 2m; lσ4 = 4m1; 2mσ2= 4m1.
We remark that the invariantslσ2 andmσ2 are even numbers.
The moduli space of K3 surfaces carrying a non-symplectic automorphism of order 8 with a given action on the K3 lattice is known to be the quotient of a complex ball of dimensionm1−1, see [8,§11],
B={[w]∈P(V) : (w, w)>0},
whereV is theζ8-eigenspace ofσ∗inT(σ4)⊗C. This implies that the Picard group of aK3 surface, corresponding to the generic point, equalsS(σ4) (see [8, Theorem 11.2]).
By [11, Theorem 3.1] the eigenvalues ofσonTX are primitive 8th roots of unity so rk(TX) = 4m1. Since 0<rk(TX)≤21 we have thatm1≤5.
2. The fixed locus
We denote by Fix(σj), j = 1,2,4 the fixed locus of the automorphism σj such that
Fix(σj) ={x∈X|σj(x) =x}.
Clearly Fix(σ) ⊆ Fix(σ2) ⊆ Fix(σ4). To describe the fixed locus of order 8 non-symplectic automorphisms we start recalling the following result about non- symplectic involutions (see [12, Theorem 4.2.2]).
Theorem 2.1. Letτ be a non-symplectic involution on aK3surfaceX. The fixed locus of τ is either empty, the disjoint union of two elliptic curves or the disjoint union of a smooth curve of genusg≥0 andksmooth rational curves.
Moreover, its fixed latticeS(τ)⊂Pic(X) is a 2-elementary lattice with determi- nant 2a such that:
• S(τ)∼=U(2)⊕E8(2)iff the fixed locus ofτ is empty;
• S(τ)∼=U⊕E8(2)iffτ fixes two elliptic curves;
• 2g= 22−rkS(τ)−aand2k=rkS(τ)−a otherwise.
The action ofσat a point in Fix(σ) can be locally diagonalized as follows (up to permutation of the coordinates, but this does not play any role in the classification):
A1,0=
ζ8 0 0 1
, A2,7=
i 0 0 ζ87
, A3,6=
ζ83 0 0 −i
, A4,5=
−1 0 0 ζ85
. In the first case the point belongs to a smooth fixed curve, while in the other cases it is an isolated fixed point. We say that an isolated pointx∈Fix(σ) is oftype(t, s) if the local action atxis given by At,s. We denote bynt,s the number of isolated fixed points byσ with matrixAt,s. We further denote by Nσj, kσj, j= 1,2,4 the number of isolated points and smooth rational curves in Fix(σj). We observe that
Nσ4 = 0 sinceσ4only fixes curves (or is empty) as explained in Theorem 2.1. For simplicity we write N :=Nσ and k:=kσ. The fixed locus of an automorphism σ is then
Fix(σ) =C∪R1∪. . .∪Rk∪ {p1, . . . , pN}
where C is a smooth curve of genusg ≥0,Ri are smooth disjoint rational curves and pi are isolated points. We will see that the fixed locus can never contain two elliptic curves or be empty.
Proposition 2.2. Letσbe a purely non-symplectic automorphism of order 8 acting on a K3 surface X. Then Fix(σ) is never empty nor it can be the union of two smooth elliptic curves. It the disjoint union of smooth curves and N ≥2 isolated points. Moreover, the following relations hold:
n2,7+n3,6= 2 + 4α, n4,5+n2,7−n3,6= 2 + 2α, N = 2 +r−l−2α.
Here we denote α= P
K⊂Fix(σ)
(1−g(K)).
Proof. Letσbe a purely non-symplectic automorphism of order 8, thenσ∗(ωX) = ζ8ωX whereζ8=e2πi8 . The Lefschetz number ofσis :
L(σ) =
2
X
j=0
(−1)j(tr(σ∗|Hj(X,OX)) = 1 +ζ87,
By [6, Theorem 4.6] we have also L(σ) =X
t,s
nt,s
det(I−σ∗|Tx)+ 1 +ζ8
(1−ζ8)2 X
C⊂Fix(σ)
(1−g(C))
where Tx denotes the tangent space at an isolated fixed point x. Comparing the two expressions for L(σ) and since L(σ) 6= 0 the fixed locus is never empty nor it can be the union of two elliptic curves. Then using the expression for the local action ofσat a fixed point we get the equations:
n2,7+n3,6 = 2 + 4α.
n4,5+n2,7−n3,6 = 2 + 2α.
(1)
Sinceα≥0 we observe thatσfixes at least two isolated points. We consider now the topological Lefschetz fixed point formula for σ. Recall that we write r =rσ
andl=lσ. We have N+P
K⊂Fix(σ)(2−2g(K)) =χ(Fix(σ)) =P4
h=0(−1)htr(σ∗|Hh(X,R))
= 2 + tr(σ∗|H2(X,R)).
This givesN+ 2α=χ(Fix(σ)) =r−l+ 2, so thatr−l=N+ 2α−2 and we have
the third relation in the statement.
Remark 2.3. The isolated fixed points, by a non-symplectic ordereightautomor- phismσof type(2,7)and(3,6)are also isolated fixed points inFix(σ2). The points of type (4,5) in Fix(σ) are contained in a smooth fixed curve by σ2. In fact the action of σ2 at such points is given by the matrix
1 0 0 ζ82
which implies that these points belong to a smooth curve inFix(σ2).
From now on we denote bynt=nt,sthe number of isolated fixed points byσof type (t, s), where t= 2,3,4 andt+s= 9. Recall the usefull lemma.
Lemma 2.4. [7, Lemma 8.1], [4] Let τ = P
iRi be a tree of smooth rational curves on a K3 surface X such that each Ri is invariant under the action of a purely non-symplectic automorphism σof orderq. Then, the points of intersection of the rational curvesRi are fixed byσand the action at one fixed point determines the action on the whole tree.
Remark 2.5. In the case of an automorphism of order 8, with the assumptions of Lemma 2.4, the local actions at the intersection points of the curves Ri appear in the following order (we give only the exponents of ζ8 in the matrix of the local action):
. . . ,(0,1),(7,2),(6,3),(5,4),(4,5),(3,6),(2,7),(1,0), . . .
Assuming that τ = R consists only of one rational curve, which is not pointwise fixed and do not intersect a fixed curve of higher genus, one get immediately that σ has either one fixed point of type (2,7) and another one of type(3,6), two fixed points of type (4,5)or one fixed point of type (4,5) and one of type(3,6).
Proposition 2.6. Let σ be a non-symplectic automorphism of order 8 on a K3 surfaceX. Assume thatPic(X) =S(σ4)andC⊂Fix(σ)withg(C) = 1. Then the following relations hold:
Nσ2 = 2kσ2+ 4,
4kσ2 = rσ2−lσ2−2 = 2(10−lσ2−mσ2).
Proof. Observe that the fixed locus ofσ2is the disjoint union ofkσ2smooth rational curves, a smooth elliptic curve andNσ2 isolated points. Moreover we have that an isolated fixed point by σ2 is given by the local action
−i 0 0 −1
. We obtain the relations in the statement by applying holomorphic and topological Lefschetz’s
formulas.
3. The classification
Lemma 3.1. If a K3 surface X carries a σ-invariant elliptic fibration, such that σ4 fixes an irreducible smooth fiberC of this fibration, thenσ acts with order 8 on the basis of the fibration and fixes two points on it.
Proof. LetπC:X −→P1 be aσ−invariant elliptic fibration havingC as a smooth fiber and such thatCis fixed by the involutionσ4. Observe thatσ2(respectivelyσ4) is not the identity on the basis ofπC, since otherwise it would act as the identity on the tangent space at a point of C, contradicting the fact thatσ2 (respectively σ4) is purely non-symplectic. Henceσ acts as an order eight automorphism onP1 and has two fixed points on it corresponding toC and another fiberC0. Remark 3.2. Let Cbe an elliptic curve fixed by the involutionσ4. ThenCis also invariant by σi fori= 1,2. Andσ either fixes C or it acts on C as a translation with no-fixed points or σ preserves C and fixes isolated points on it. In this last case we have two possibilities:
a) The automorphism σ acts on C as an automorphism of order four with two isolated fixed points, that must be of type (2,7) and/or (3,6). In fact σ can not have a fixed point of type(4,5) onC otherwise this point would be contained on a fixed curve forσ2, that would be also fixed byσ4, butσ4 already fixes C and fixed curves do not intersect.
b) The automorphism σ acts on C as an involution with four isolated fixed points. These must be clearly of type (4,5).
Theorem 3.3. Let σ be a purely non-symplectic automorphism of order 8 on a K3 surfaceX, such that the involution σ4 fixes a smooth elliptic curve. Then the number#C of fixed smooth elliptic curves byσ4 is at most2. Moreover, the action of σis described in the Table 5.
Proof. Observe first that rk Pic(X) ∈ {18,14,10,6,2} since ϕ(8)|rkTX by [11].
The involutionσ4 fixes an elliptic curve by Theorem 2.1 (see also [5, Figure 1]) so that we get rk Pic(X)∈ {10,14,18}. On the other hand, by Theorem 2.1 again we have that σ4 fixes at most two elliptic curves, moreover we know that for the number #C of smooth fixed elliptic curves byσ4 and the number kσ4 of rational fixed curves holds:
(2) (rk Pic(X),#C, kσ4) = (10,2,0),(14,1,4),(18,1,8).
Let C denote a σ4-fixed smooth elliptic curve. Since σ preserves C there is a σ-invariant elliptic fibration
πC:X−→P1
with generic fiber C. Observe that by Lemma 3.1 the automorphismσ has order eight on the basisP1and it fixes two points corresponding to the fiberCand a fiber C0. The fiber C0 can be smooth elliptic and sokσ = 0 or it can be reducible and then it contains all rational curves fixed byσ4. In the case (rk Pic(X),#C, kσ4) = (14,1,4), the curveC0 is a reducible fiber containing four smooth rational curves fixed byσ4. Since the non-symplectic involutionσ4does not fix isolated points and the fixed curves by σ4 do not intersect, one can easily see that the reducible fiber C0 is either of Kodaira typeIV∗ or I8. By the same argument we get thatC0 is of type I16 in the case (rk Pic(X),#C, kσ4) = (18,1,8) (see also [4, Theorem 3.1, Theorem 8.4]).
The caserk Pic(X) = 10. Since kσ4 = 0 we have alsok= 0, thus n2+n3= 2 by Proposition 2.2. Observe that σacts as an automorphism of order four on one of the two σ4-fixed elliptic curves C, C0, in fact σ2 has order four and can not fix two elliptic curves, see [4, Proposition 1]. On the other hand either σ acts on the other σ4−fixed elliptic curve as a translation (of order 2 or 4, recall that the fixed locus ofσ4 consists of two smooth eliptic curves) or it is the identity, in this case n4 = 0 so that (n2, n3, n4) = (2,0,0) or it acts as an involution when (n2, n3, n4) = (0,2,4) (see Proposition 2.2 and Remark 3.2). Since rk Pic(X) = 10 we have that rkTX = 12 so that m1 = 3. This gives mσ2 = 6 by Remark 1.1.
Since kσ2 = 0 using Proposition 2.6 we get that 10−lσ2−mσ2 = 0, so we have lσ2 = 2m = 4, m = 2 andrσ2 = 6 by Proposition 2.6 again. By Proposition 2.2 and Remark 1.1 one can find easily that (r, l, m) = (3,3,2), respectively (5,1,2).
This gives thefirst four cases in Table 5.
The caserk Pic(X) = 14. As we have already seen, the fiberC0is either of type IV∗ orI8. We study here these two possibilities separately.
• C0 of typeIV∗: Since the fiber IV∗ is preserved by σi ; i = 1,2,4, the automorphismσeither preserves each component ofIV∗or it exchanges the two branches ofIV∗. In these two casesσ2preserves each component ofC0 and fixes the central component of multiplicity 3 since it has at least three fixed points byσ2. Then we have thatk∈ {0,1} by Remark 2.5. Ifk= 1 thenσpreserves each component ofC0, hence it fixes six points onC0 three of each type (2,7) and (3,6) by Remark 2.5, thus (n2, n3, n4)≥(3,3,0) (we write (x1, x2, x3)≥(y1, y2, y3) if and only if xi≥yi for alli= 1,2,3.). By Proposition 2.2 we get (n2, n3, n4) = (3,3,4), this implies thatσacts as an involution on C and fixes four points on it of type (4,5). This iscase 11 in Table 5.
If k = 0 the automorphism σ exchanges the two branches of C0. So the fiberC0 contains two fixed points of type (4,5) on the central component of multiplicity 3, which is then fixed by σ2, and one point of type (2,7) and another one of type (3,6), hence we get (n2, n3, n4) ≥ (1,1,2). By Proposition 2.2 we have that (n2, n3, n4) = (1,1,2) which means that σ acts on C either as the identity or as a translation of order two or four (sinceσ4 fixesC). These are thecases 5,6,7 in the Table 5.
Finally, sincekσ2 = 1 and by doing the same computation as in the previous case one gets immediately that (r, l, m) = (10,0,2) and (6,4,2) respectively fork= 1 and k= 0.
• C0 of typeI8: First observe that all components ofC0 are preserved byσ4 since Pic(X) = S(σ4) and a component which is not fixed intersects two fixed ones. The automorphism σ either preserves each component of I8
or acts on it as a reflection (i.e. σ preserves two components on I8 and exchanges the remaining 6 components two by two) or it acts as a rotation, of order two or four. Applying Remark 2.5 one gets that in the first caseσ fixes one component of C0 (i.e. k = 1) and 2 isolated points of each type (2,7),(3,6) and (4,5), so we get (n2, n3, n4)≥(2,2,2). On the other hand, we have that (n2, n3, n4) = (4,2,2) by Proposition 2.2, which implies that σacts onCas an automorphism of order four and fixes two points on it of type (2,7) (see Remark 3.2). This is thecase 12in the Table 5.
If σ acts on C0 as a reflection, then k = 0 and σ fixes four points of type (4,5) on the two invariants components ofC0. Observe that these two components are fixed byσ2. Hence we get (n2, n3, n4)≥(0,0,4). On the other hand, by Proposition 2.2 we have that (n2, n3, n4) = (0,2,4), thus σ acts as an automorphism of order four on C and fixes two points of type (3,6) (see Remark 3.2). This is thecase 10in the Table 5.
Using the relation 4kσ2 =rσ2 −lσ2−2 = 2(10−lσ2 −mσ2) of Propo- sition 2.6 since kσ2 = 2 and mσ2 = 4 we get in these two cases that (rσ2, lσ2) = (12,2). In fact in these two casesσ2preserves each component of I8 and fixes two components in C0. We get then easily that (r, l, m) is equal respectively to (10,2,1) and (8,4,1) by Proposition 2.2.
Finally, if σ acts as a rotation of order two or of order four on C0, then in these both cases k = 0 and σ does not fix any point on C0. By Proposition 2.2 we have that (n2, n3, n4) = (2,0,0) and so σ also acts as an automorphism of order four on C and fixes two points of type (2,7) (see Remark 3.2). Moreover we compute as above the invariants r, l, m in
the case thatkσ2 = 2 respectively 0 which correspond to the action of the automorphism onI8as a rotation of order two or of order four respectively.
We get easily that (r, l, m) equals (6,6,1) respectively (4,4,3). These are thecases 8 and 9in the Table 5.
The caserk Pic(X) = 18. In this case the fiberC0is of typeI16as remarked at the beginning of the proof. As we have seen previously, there are four possibilities for the action ofσonI16 that are: eitherσpreserves each component ofI16or it acts as a reflection (hereσpreserves two components ofI16and exchanges the remaining 14 components two by two) or it acts as a rotation of order two, respectively four.
IThese are the cases 16, 15, 13 and 14 in the Table 5. If σ preserves each component ofC0, then by applying Remark 2.5 we get thatσfixes two components of C0 (i.e. k = 2) and 12 isolated points four of each type (2,7),(3,6) and (4,5).
On the other hand, if σ acts as a reflection, then k = 0 and σ fixes four isolated points of type (4,5) on the two preserved components ofC0. Finally,σdoes not fix any point onC0 if it acts as a rotation.
Observe that in these three casesσ acts as an automorphism of order four on the smoothσ4−fixed fiber C by Proposition 2.2. Using the same argument as before we get the values of the invariants r, l, m that appeared in the last four cases of
Table 5.
4. Examples
In this section we give examples corresponding to several cases in the classifica- tion of the non-symplectic automorphisms of order eight on ellipticK3 surfaces.
Example 4.1. Consider the elliptic fibration πC : X −→ P1 with a Weierstrass equation:
y2=x3+a(t)x+b(t).
wherea(t) =at8+b andb(t) =ct8+dwitha, b, c, d∈C. The fibrationπC admits the order eight automorphism:
σ(x, y, t) = (x, y, ζ8t).
The fibers preserved byσ are over 0 and∞and the action ofσat infinity is:
(x/t4, y/t6,1/t)7−→(−x/t4, iy/t6, ζ87/t) The discriminant polynomial ofπC is:
∆(t) := 4a(t)3+ 27b(t)2=h1t24+h2t16+h3t8+O(t4), where
h1= 4a3, h2= 12a2b+ 27c2, h3= 12ab2+ 54cd.
Observe that ∆(t) has 24 simple zeros for a generic choice of the coefficients. By studying the zeros of ∆(t) and looking in the classification of singular fibers of elliptic fibrations on surfaces (see e.g. [10, section 3]) one obtains the following: for a generic choice of the coefficients ofa(t) andb(t) the fibration has 24 fibers of type I1 over the zeros of ∆(t), both fibers over t = 0 and t = ∞ are smooth elliptic curves, moreover the automorphism σ fixes pointwisely the fiber over 0 and acts as anorder 4 automorphism on the fiber over∞, hence it has two fixed points on this fiber. We show below that generically the rank of Pic(X) is 10. This gives an example for thefirst case in Table 5. On the other hand, ifh1 = 0 so that a = 0 the fibration acquires a fiber of type IV∗ at ∞ by a generic choice of the
parameters. The fiber over 0 is a smooth elliptic curve fixed pointwisely byσ. This gives an example for the case 5 in Table 5, this follows by Theorem 3.2 and by the fact that this case is the only one for which σcontains an elliptic curve in the fixed locus and the fibration has a fiber of typeIV∗.
By using standard transformations on the parameters in the Weierstrass form we get that the number of moduli is 2. In fact both the polynomialsa(t), b(t) depend on 2 parameters, but we can apply the transformation (x, y)7→(λ2x, λ3y) ; λ∈C∗, to cancel one of the 4 parameters. Moreover the automorphisms ofP1 commuting witht7→ζ8tare of the formt7→µt; µ∈C∗, so we can cancel a second parameter.
This shows that the family depends on 2 parameters, so that generically rkTX = 12 (recall that rkTX = 4m1 and m1−1 equals to the number of moduli) and rk Pic(X) = 10.
Example 4.2. As in the previous example 4.1, consider again the elliptic fibration πC:X −→P1 in Weierstrass form given by :
y2=x3+a(t)x+b(t).
where a(t) =at8+b and b(t) =ct8+d with a, b, c, d∈ C. This elliptic fibration carries the non-symplectic automorphismσof order eight:
(x, y, t)7→(x,−y, ζ8t).
The fibers preserved byσ are over 0,∞and the action at infinity is (x/t4, y/t6,1/t)7−→(−x/t4,−iy/t6, ζ87/t).
The discriminant polynomial ofπC is:
∆(t) := 4a(t)3+ 27b(t)2=h1t24+h2t16+O(t8), where
h1= 4a3 andh2= 12a2b+ 27c2.
We have seen in example 4.1 that for a generic choice of the coefficients ofa(t) andb(t) the fibration has 24 fibers of type I1over the zeros of ∆(t) (see [10, section 3]), moreoverσ acts as aninvolutionon the fiber over 0 and it acts as anorder 4 automorphismon the fiber over∞(both fibers are smooth). So we have an example for the case 4 in Table 5. Ifh1= 0, that impliesa= 0, the fibration acquires a fiber of type IV∗at∞by a generic choice of the parameters. This is the case 11 in Table 5 (this is the only case where the fibartion has a fiber of type IV∗ and the action on the smooth fiber is an involution).
Example 4.3. Consider the elliptic fibration πC :X −→P1 in Weierstrass form given by
y2=x3+a(t)x+b(t),
where a(t) =at8+b and b(t) =ct4+dt12 ; a, b, c, d ∈C. Observe that it carries the order eight automorphism
σ: (x, y, t)7−→(−x, iy, ζ87t).
For generic choice of the coefficients the fiber overt= 0 is smooth. The automor- phismσ4 is an involution fixing the smooth elliptic curve overt= 0. On the other hand, σacts on the elliptic curve over t = 0 as an order 4 automorphism with 2 isolated fixed points. Moreover it acts as the identity on the fiber over t=∞, in fact the action is
(x/t4, y/t6,1/t)7−→(x/t4, y/t6, ζ8/t).
The discriminant ofπC is :
∆(t) =h1t24+h2t16+h3t8+ 4b3, where
h1= 4a3+ 27d2 , h2= 12a2b+ 54cd , h3= 12ab2+ 27c2.
Observe that ∆(t) has 24 simple zeros for a generic choice of the coefficients. By studying the zeros of ∆(t) and looking in the classification of singular fibers of elliptic fibrations on surfaces (e.g [10, section 3]) one obtains the following cases:
for the generic choice of the coefficients ofa(t) andb(t) the fibration has 24 fibers of typeI1. We get again an example forcase 4 in Table 5. Ifh1= 0 the fibration has a fiber of type I8 at infinity. If h1 =h2 = 0 we get a fiber I16. By [9,§3] a holomorphic two form is given byωX = (dt∧dx)/2yand so the action ofσon it is σ∗(ωX) = −ζi78ωX =ζ8ωX. We determine now exactly the type of the local action ofσat the two fixed points on the elliptic curveC. We look at the elliptic fibration locally around the fiber overt= 0. The equation inP2×Cis given by:
F(x, y, z, t) :=zy2−(x3+ (at8+b)z2x+ (ct4+dt12)z3) = 0.
Where (x:y:z) are the homogeneous coordinates ofP2 . The fiber at t= 0 has equation
f :={zy2−x3−bz2x= 0}
and forb∈Cgeneric is smooth since the partial derivatives∂f /∂x, ∂f /∂y, ∂f /∂z are not simultaneously zero. The two fixed points by σ are p := (0 : 1 : 0) and p0 := (0 : 0 : 1). The pointp0 := (0 : 0 : 1) is contained in the chartz = 1 and it belongs to the open set Fx := ∂F/∂x6= 0, in fact Fx(p0) := ∂F(x,y,1,0)
∂x 6= 0. The one-form for the elliptic curve in this open subset is:
dy/Fx(p0) =dy/(−3x2−b).
The action ofσ here is a multiplication byi: (σ∗(dy/Fx(p0)) = i(dy/Fx(p0))), so that the action on the holomorphic 2-form dt∧(dy/(−3x2(at8+b)) is the multi- plication by ζ8 as expected. In particular the local action at p0 is of type (7,2).
Similarly we can do the computation on the open subset in the charty = 1 which contains the fixed pointp, and we can find again the same action (7,2). Observe that since σ acts as the identity on the fiber over t =∞, it preserves the curves of the singular fibers (of typeI8 orI16). This gives an example for the cases 12 and 16 in Table 5.
On other hand, the fibration πC admits also the automorphism τ(x, y, t) = (−x,−iy, ζ83t). This automorphism acts also by multiplication by ζ8 on the holo- morphic 2-form ωX, thus τ is not a power of σ (i.e it is a new automorphism).
Moreover the square ofτ preserves each components of the fiber att=∞ in fact the action at infinity is:
(x/t4, y/t6,1/t)7−→(x/t4,−y/t6, ζ85/t).
By a similar computation as above one sees that the local action at the fixed points on the fiber C is of type (3,6), so we have an example for the cases 10 and 15 in Table 5respectively.
Example 4.4. (Translation).
We give here an example for the cases 2 , 8 and 13 in Table 5. Observe that
in these three cases the non-symplectic automorphism of order 8 on X acts as a translation of order two on theσ-invariant fiberC0.
It is well known that an ellipticK3 surface with a 2-torsion section can be written with equation:
y2=x(x2+a(t)x+b(t),
wherea(t) andb(t) are polynomials of degree 4 and 8 respectively. Or equivalently one can write the equation in Weierstrass form:
y2=x3+A(t)x+B(t), where
A(t) = 9b(t)−3a(t)2 andB(t) = 3a(t)2−9a(t)b(t).
Now the map:
(4) τ: (x, y, t)7→(y2/x2−a(t)−x,(y/x).τ(x), t),
withτ(x) := (y2/x2)−a(t)−xis an automorphism onX that acts as a translation of order two on the generic fiber ofπ.
Consider now the non-symplectic automorphism of order eight onX:
σ: (x, y, t)7→(−x, iy, ζ87t).
This automorphism preserves the jacobian elliptic fibrationπ :X −→ P1 defined as follows:
y2=x(x2+a(t)x+b(t)),
wherea(t) =αt4, b(t) =βt8+γ; α, β, γ∈C. Or equivalently the fibration can be written as:
y2=x3+A(t)x+B(t),
such that A(t) = (9β −3α2)t8+ 9γ and B(t) = (2α3 −9αβ)t12−9αγt4. The discriminant ofπis:
∆(t) =K(βt8+γ)2[(α2−4β)t8−4γ]; K∈Cis a constant.
Consider now the translationτ :
τ : (x, y, t)7→(y2/x2−αt4−x,(y/x).τ(x), t).
As we have seen,τis an automorphism ofXand it acts as a translation of order two on the generic fiber ofπ. Moreover, one can get easily thatτ◦σ=σ◦τ, thus theK3 surface X has the order eight non-symplectic automorphism σ0 :=σ◦τ. Observe that the automorphismsσandσ0 act with order eight onP1, they preserve the two fibers overt= 0 andt=∞and act as an automorphism of order four on the smooth fiber overt= 0 given byf :=zy2−x3−9γxz2, forγ∈Cgeneric. Moreover,σacts as the identity on the fiber overt=∞, whileσ0acts on it as an order two translation (observe that the action ofσat infinity is (x/t4, y/t6,1/t)7→(x/t4, y/t6, ζ81/t) and σ02= (σ◦τ)2=id).
Studying the zeros of the discriminant ∆(t), and looking in the classification of singular fibers of elliptic fibrations on surfaces (e.g [10,§3]) we get the following:
• For generic α, β, γ the fibration π has 8 fibers of typeI2 and 8 fibers of typeI1 over the zeros of ∆(t), σ0 acts as an order four automorphism on the fiber over 0 and it acts as an order two translation on the fiber at ∞ (both fibers are smooth). This gives an example for thecase 2 in Table 5.
• Ifα2−4β = 0 with (β6= 0), then ∆(t) =K(βt8+γ)2(−4β) ; K∈C∗.So that the fibration acquires a fiber of typeI8over∞(in fact ∆(t) has a zero of order 8 overt=∞andA(t), B(t) are non zero). The automorphism σ0 acts as a rotation of order two on the fiberI8 (see [10]). This corresponds to thecase 8 in Table 5.
• If β = 0, (α 6= 0), the discriminant ∆(t) = Kγ2(α2t2−4γ) vanishes at t=∞with order 16, andA(t), B(t) are nonzero at∞. Thus we get a fiber of typeI16, on whichσacts as a rotation of order two. We are in thecase 13 of Table 5.
Example 4.5. Quadruple Quartics.
Take the fourfold cover ofP2:
t4=x0(l3(x1, x2) +x20l1(x1, x2))
where l3(x1, x2) is homogeneous of degree three and l1(x1, x2) is homogeneous of degree 1. This is invariant for the action of the order 8 non symplectic automor- phism:
(t, x0, x1, x2)7→(ζ8t,−x0, x1, x2)
it fixes the inverse image of the curve{x0 = 0} which is rational and 4 points on the curve C : {l3(x1, x2) +x20l1(x1, x2) = 0} which is in fact elliptic. This gives another example for thecase 11 of Table 5.
5. The Table of the σ-invariant elliptic fibrations
We give the table for the classification of the non-symplectic automorphisms of order 8 on an elliptic K3 surface. The cases for which we have an example are denoted with√
. We list also in this table the invariantsr, l, mandm1 ofσwhich denote the rank of the eigenspaces of (σ)∗ in H2(X,C) relative to the eigenvalues 1,−1, iandζ8respectively. Recall that by #Cwe denote the number ofσ-invariant smooth elliptic curves.
Examplerlmkσ2#CrkPic(X)kσ4N(n2,n3,n4)kactionofσonfixedellipticcurvesandsingularfibers1. √332021002(2,0,0)0(identity,orderfour) 2. √33202(2,0,0)0(translationofordertwo,orderfour)
3.−33202(2,0,0)0(translationoforderfour,orderfour)
4. √51206(0,2,4)0(involution,orderfour) 5. √642111444(1,1,2)0(identity,reflectionofIV∗) 6.−64214(1,1,2)0(translationofordertwo,reflectionofIV∗) 7.−64214(1,1,2)0(translationoforderfour,reflectionofIV ∗) 8. √66122(2,0,0)0(orderfour,rotationoforder2onI8) 9.−44302(2,0,0)0(orderfour,rotationoforder4onI8) 10. √84126(0,2,4)0(orderfour,reflectiononI8) 11. √1002110(3,3,4)1(involution,preserveseachcurveofIV ∗) 12. √102128(4,2,2)1(orderfour,preserveseachcurveofI8) 13. √990411882(2,0,0)0(orderfour,rotationoforder2onI16) 14.−55402(2,0,0)0(orderfour,rotationoforder4onI16) 15. √117046(0,2,4)0(orderfour,reflectiononI16) 16. √1710414(6,4,4)2(orderfour,preserveseachcurveofI16)
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Laboratoire de Math´ematiques et Applications, UMR CNRS 7348, Universit´e de Poitiers, T´el´eport 2, Boulevard Marie et Pierre Curie, 86962 FUTUROSCOPE CHASSENEUIL, France
E-mail address:[email protected]
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